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By Newton's second law, the net inward force on an object in circular motion must equal mv²/r (the centripetal force). 'Centripetal force' is not a new type of force — it is the net component of real forces (tension, gravity, normal force, friction) directed toward the center. Setting ΣF_inward = mv²/r is the governing equation for circular dynamics problems.
Draw a free-body diagram, identify which forces point toward the center, and set their net inward component equal to mv²/r. Vertical circles (roller coasters, buckets of water) require analyzing the top and bottom of the loop separately.
When you first studied Newton's second law, you applied it to objects moving in straight lines: ΣF = ma, where acceleration is in the same direction as the net force. Circular motion adds one new insight: an object moving in a circle is constantly accelerating even if its speed is constant, because its direction is changing. From circular motion kinematics, you know this centripetal acceleration points toward the center and has magnitude v²/r. Plugging this into Newton's second law gives ΣF_inward = mv²/r — the fundamental equation of circular dynamics.
The most important conceptual point is that "centripetal force" is not a new type of force that you add to free-body diagrams. It is a label for the net inward component of whatever real forces happen to be present: tension in a string, gravity, normal force, friction, or any combination. When solving circular motion problems, your job is to draw the free-body diagram with only real forces, identify which components point toward the center, sum them, and set that sum equal to mv²/r.
Vertical circle problems — roller coasters, buckets of water swung overhead, cars over hills — require extra care because the direction of "inward" changes around the loop. At the bottom of a loop, inward means upward, so N − mg = mv²/r (normal force exceeds weight, which is why you feel heavy at the bottom of a roller coaster dip). At the top of a loop, inward means downward, so both mg and N point inward: mg + N = mv²/r. This is why you feel lighter at the top. The minimum speed at the top of a loop corresponds to N = 0; below that speed, the track would have to pull rather than push, which it cannot do, and the car leaves the track.
A persistent error is adding a fictitious "centrifugal force" outward to the free-body diagram — perhaps because you can feel yourself pushed outward when riding in a turning car. That sensation is real, but it is your body's inertia resisting the inward acceleration, not an outward force acting on you. In the inertial reference frame where Newton's laws hold without modification, no centrifugal force exists. In a rotating reference frame, centrifugal force is a valid pseudo-force, but mixing the two frames in one diagram produces wrong answers.